The question asks to calculate $ \vec{ez}$ in function of $\vec{i}$ $\vec{j}$ and $\vec{k}$.
$\alpha=(Ox,Ox_1)=(Oz,Oz_1)$
$\vec{i}$ $\vec{j}$ and $\vec{k}$ being respectively the unit vectors of $OX1$, $OY$ and $OZ1$
$\vec{ez}$ is the unit vector of $Oz$
According to the solution manual : $ \vec{ez}=cos(\alpha) \vec{k}-sin(\alpha) \vec{i}$
However i'm not sure how they got to this answer.
Shouldn't it be $ \vec{ez}=cos(\alpha) \vec{k}$ instead?
Thanks in advance and excuse me for the bad drawing!
Note that $z_1-x_1$ seems obtained by a rotation of $z-x$ orthogonal to $y$ then
$$\vec{e_z}=cos(\alpha) \vec{k}-sin(\alpha) \vec{i}$$
is correct and
$$\vec{e_x}=sin(\alpha) \vec{k}+cos(\alpha) \vec{i}$$
note indeed that $$\vec{e_z}\cdot \vec{e_x}=0$$